Olivier’s theorem: ideal convergence, algebrability and Borel classification

The classical Olivier’s theorem says that for any nonincreasing summable sequence (a(n)) the sequence (na(n)) tends to zero. This result was generalized by many authors. We propose its further generalization which implies known results. Next we consider the subset $${\mathcal {AOS}}$$ of $$ \ell _{1} $$ consisting of sequences for which the assertion of Olivier’s theorem is false. We study how large and good algebraic structures are contained in $${\mathcal {AOS}}$$ and its subsets; this kind of study is known as lineability. Finally we show that $${\mathcal {AOS}}$$ is a residual $$ \mathcal {G}_{\delta \sigma } $$ but not an $$ {\mathcal {F}}_{\sigma \delta } \text {-set} $$ .